The average velocity of a particle in a time interval t_{1} to t_{2} is defined as its displacement divided by the time interval.

For example – If the particle is at a point A at time t = t_{1} and at B at time t = t_{2}

The displacement in this time interval is the vector AB.

Then average velocity in this time interval is

Like displacement, average velocity is also a Vector quantity.

Moreover we can further simplify this equation as from triangle AOB it’s clear that

It’s clear from above equation that average velocity takes into consideration only final and initial positions of object and doesn’t care about how object went from initial to final position.

Let’s understand this using an example problem.

ProblemA table clock has it minute hand 4.0 cm long. Find the average velocity of the tip of the minute hand. 1) Between 6:00 am to 6:30 am 2) Between 6:00 am to 6:30 pm |

Solution1) Between 6:00 am to 6:30 am At 6:00 am minute hand of clock is at 12 mark on clock At 6:30 am minute hand of clock is at 6 mark on clock |

Displacement = 8.0 cm Time Taken = (from 6:00 am to 6:30 am is 30 minutes) = 30 min = 30 × 60 = 1800 s Time Taken = 1800 s Average velocity = v _{av} = Displacement/Time Taken = 8.0 / 1800 = 4.4 × 10^{-3} cm/s |

Solution2) Between 6:00 am to 6:30 pm At 6:00 am minute hand of clock is at 12 mark on clock At 6:30 am minute hand of clock is at 6 mark on clock Similar to above question, displacement of minute hand in this case will be 8.0 cm Displacement = 8.0 cm But in this case time has gone from 6:00 am to 6:30 pm which is 12 hours 30 minutes which is 45000 seconds. Displacement = 8.0 cm Time Taken = 45000 s Average velocity = v _{av} = Displacement/Time Taken = 8.0 / 45000= 1.8 × 10 ^{-4} cm/s⇒ Average velocity = v _{av} = 1.8 × 10^{-4} cm/s |

Let’s now discuss Instantaneous Velocity

Average Velocity of the particle in a short time interval t to t + Δt is defined as follows.

where Δr vector is the displacement in the time interval Δt

Now if we make Δt vanishingly small and find the limiting value of Δr/Δt

This value is Instantaneous Velocity of the particle at time t.

Let’s understand Instantaneous Velocity using an example.

ProblemLet’s suppose that a particle is moving on a path and it’s displacement function is given as 5t ^{2}.Then find out Instantaneous Velocity of this particle at t = 10 s |

SolutionInstantaneous Velocity at a particular instance of time t is given as dr/dt r = 5t ^{2} as per questiondr/dt = d(5t ^{2})/dt = 10tThus Instantaneous Velocity function of particle is 10t At t = 10 s Instantaneous Velocity = 100 m/s |

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